---
tag: XR4DPKX7T1
kind: concept
name: Pseudovariety
msc: ["08B99", "20M07"]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

A pseudovariety is a class of finite algebras of one signature that is closed under homomorphic images, subalgebras and finite direct products.
It is the analogue of a [[variety|AJNZJD8442]] for finite algebras.
Pseudovarieties are not always defined by identities, but [[Reiterman's theorem|NT8W7R2KDR]] shows that they are defined by pseudoidentities.
Pseudovarieties of finite semigroups and monoids classify regular languages in algebraic automata theory.

## Explanations

### L1 human

Start from some finite algebras and allow three moves: take a quotient, take a subalgebra, or take the product of finitely many.
A class that is stable under these moves is a pseudovariety.
Finite groups, finite abelian groups and finite nilpotent groups are examples.
Finite groups regarded as semigroups also form a pseudovariety, yet no set of semigroup identities captures exactly the finite groups among finite semigroups.
The reason is that the infinite semigroup $(\set{1, 2, 3, \ldots}, +)$ lies in every variety of semigroups containing all finite groups, and it has finite quotients that are not groups.